arXiv subjects

Yuanyuan Luo

Publications and source records attributed to Yuanyuan Luo.

4 recordsLinked to original sources

arXiv2025

Toric structure of the moduli space of points in projective space

Gallardo and Routis constructed compactifications of the moduli space of $n$ labeled points in $\mathbb{P}^d$ by assigning weights to points, generalizing Hassett's weighted compactifications of $M_{0,n}$ to higher-dimensional projective spaces. Among their compactifications, there is a toric compactification that generalizes the standard Losev-Manin compactification to this higher-dimensional setting. Our main result identifies the fan of this toric compactification as a symmetric product of a nested fan, generalizing the classical connection between Losev-Manin spaces and the permutohedron to arbitrary dimension. More generally, we prove that the fans of all Gallardo-Routis compactifications that admit reduction maps from this Losev-Manin space are symmetric products of building sets. This shows that the combinatorics of these compactifications are controlled by coarsenings of the permutohedral fan that give rise to Hassett spaces.

math.AG↗

arXiv2025

Factorization of power GCD matrices and power LCM matrices on certain gcd-closed sets

For integers $x$ and $y$, $(x, y)$ and $[x, y]$ stand for the greatest common divisor and the least common multiple of $x$ and $y$ respectively. Denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix having the $a$th power of $(x_i,x_j)$ (resp. $[x_i,x_j]$) as its $(i,j)$-entry. For any $x\in S$, define $G_{S}(x):=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S) \Rightarrow y\in \{d,x\}\}$. In this paper, we show that if $a|b$ and $S$ is gcd closed (namely, $(x_i, x_j)\in S$ for all integers $i$ and $j$ with $1\le i, j\le n$) and $\max_{x\in S}\{|G_S (x)|\}=3$ such that any elements $y_1,y_2\in G_S(x)$ satisfy that $[y_1,y_2]=x$ and $(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$), then $(S^a)|(S^b)$, $(S^a)|[S^b]$ and $[S^a]|[S^b]$ hold in the ring $M_{n}({\mathbb Z})$. This extends the Chen-Hong-Zhao theorem gotten in 2022. This also partially confirms a conjecture of Hong raised in [S.F. Hong, Divisibility among power GCD matrices and power LCM matrices, {\it Bull. Aust. Math. Soc.}, doi:10.1017/S0004972725100361].

math.NT↗

arXiv2014

The elementary symmetric functions of a reciprocal polynomial sequence

Erdös and Niven proved in 1946 that for any positive integers $m$ and $d$, there are at most finitely many integers $n$ for which at least one of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ are integers. Recently, Wang and Hong refined this result by showing that if $n\geq 4$, then none of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ is an integer for any positive integers $m$ and $d$. Let $f$ be a polynomial of degree at least $2$ and of nonnegative integer coefficients. In this paper, we show that none of the elementary symmetric functions of $1/f(1), 1/f(2), ..., 1/f(n)$ is an integer except for $f(x)=x^{m}$ with $m\geq2$ being an integer and $n=1$.

math.NT↗

arXiv2013

Uniform lower bound for the least common multiple of a polynomial sequence

Let $n$ be a positive integer and $f(x)$ be a polynomial with nonnegative integer coefficients. We prove that ${\rm lcm}_{\lceil n/2\rceil \le i\le n} \{f(i)\}\ge 2^n$ except that $f(x)=x$ and $n=1, 2, 3, 4, 6$ and that $f(x)=x^s$ with $s\ge 2$ being an integer and $n=1$, where $\lceil n/2\rceil$ denotes the smallest integer which is not less than $n/2$. This improves and extends the lower bounds obtained by Nair in 1982, Farhi in 2007 and Oon in 2013.

math.NT↗